基于周期变换求解带尖角区域的声波散射问题

利用周期变换和位势理论将声波散射问题转化为第二类边界积分方程,再利用Nystrom方法来求解该边界积分方程.给出二维空间的数值例子,结果表明该方法简单,可行并且具有较好的精度.     

An Exterior Differential System for a Generalized Korteweg-de Vries Equation and its Associated Integrability

The method of Cartan is reviewed by applying it to the classical Korteweg-de Vries equation. The method is then applied to a new generalized Korteweg-de Vries equation for which a prolongation is obtained. As a consequence, a Bäcklund transformation for the equation is derived as well as the associated potential equation.     

BCKLUND TRANSFORMATION AND LAX REPRESENTATION FOR A NONLINEAR DIFFERENTIAL EQUATION

In this paper, the Hirota bilinear method is applied to a nonlinear equation which is a deformation to a KdV equation with a source. Using the Hirota’s bilinear operator, we obtain its bilinear form and construct its bilinear Bcklund transformation. And then we obtain the Lax representation for the equation from the bilinear Bcklund transformation and testify the Lax representation by the compatibility condition.     

On the multiple hyperbolic systems modelling phase transformation kinetics

We discuss Cahn’s time cone method modelling phase transformation kinetics. The model equation by the time cone method is an integral equation in the space-time region. First, we reduce it to a system of hyperbolic equations, and in the case of odd spatial dimensions, the reduced system is a multiple hyperbolic equation. Next, we propose a numerical method for such a hyperbolic system. By means of alternating direction implicit methods, numerical simulations for practical forward problems are implemented with satisfactory accuracy and efficiency. In particular, in the three dimensional case, our numerical method on the basis of reduced multiple hyperbolic equation is fast.     

齐次平衡法寻找广义Caudrey-Dodd-Gibbon-Kaeada方程的多孤子解

利用齐次平衡法寻找Hirota变换,再通过Hirota变换将方程转化为Hirota双线性形式,进一步解释两种方法之间的联系,并得出将一些方程转化为Hirota双线性形式的一般步骤.     

Nizhnik方程组的一个非线性变换和多重孤子解

用齐次平衡原则导出了一个非线性变换,通过该变换Nizhnik方程组化为一个齐2次方程.用Hirota方法可求出齐2 次方程的一列解.将其代入非线性变换,得Nizhnik方程组的多重孤子解.详细分析了二重孤子解.     

N-soliton solutions for the integrable bidirectional sixth-order Sawada-Kotera equation

In this work, the integrable bidirectional sixth-order Sawada-Kotera equation is examined. The equation considered is a KdV6 equation that was derived from the fifth order Sawada-Kotera equation. Multiple soliton solutions and multiple singular soliton solutions are formally derived for this equation. The Cole-Hopf transformation method combined with the Hirota’s bilinear method are used to determine the two sets of solutions, where each set has a distinct structure.     

Exact solutions for the generalized Klein–Gordon equation via a transformation and Exp-function method and comparison with Adomian’s method

In this paper, a suitable transformation and a so-called Exp-function method are used to obtain different types of exact solutions for the generalized Klein–Gordon equation. These exact solutions are in full agreement with the previous results obtained in Refs. [Sirendaoreji, Auxiliary equation method and new solutions of Klein–Gordon equations, Chaos, Solitons & Fractals 31 (4) (2007) 943–950; Huiqun Zhang, Extended Jacobi elliptic function expansion method and its applications, Communications in Nonlinear Science and Numerical Simulation, 12 (5) (2007) 627–635]. One of these exact solutions is compared with the approximate solutions obtained by the modified decomposition method. Accurate numerical results for a wider range of time are obtained after using different types of ADM-Padè approximation. Our results show that the Exp-function method is very effective in finding exact solutions for the problem considered while the modified decomposition method is very powerful in finding numerical solutions with good accuracy for nonlinear PDE without any need for a transformation or perturbation.     

mKdV-SineGordon方程的多孤子解

该文首先给出了mKdV-SineGordon方程的双线性形式和双线性Backlund变换,然后利用Hirota方法、Backlund变换方法和Wronskian技巧三种不同的方法分别得到mKdV-SineGordon方程的孤子解,最后验证了这三种解的一致性。     

Group method solution for solving nonlinear heat diffusion problems

The linear transformation group approach is developed to simulate heat diffusion problems in a media with the thermal conductivity and the heat capacity are nonlinear and obeyed a striking power law relation, subject to nonlinear boundary conditions due to radiation exchange at the interface according to the fourth power law. The application of a one-parameter transformation group reduces the number of independent variables by one so that the governing partial differential equation with the boundary conditions reduces to an ordinary differential equation with appropriate corresponding conditions. The Runge–Kutta shooting method is used to solve the nonlinear ordinary differential equation. Different parametric studies are worked out and plotted to study the effect of heat transfer coefficient, density and radiation number on the surface temperature.     

Radial Schrödinger equation: The spectral problem

Using the integral transformation method involving the investigation of the Laplace transforms of wave functions, we find the discrete spectra of the radial Schrödinger equation with a confining power-growth potential and with the generalized nuclear Coulomb attracting potential. The problem is reduced to solving a system of linear algebraic equations approximately. We give the results of calculating the discrete spectra of the S-states for the Schrödinger equation with a linearly growing confining potential and the nuclear Yukawa potential.     

Inverse scattering transformation and soliton stability for a nonlinear Gross–Pitaevskii equation with external potentials

We consider the Gross–Pitaevskii(GP) equation with the combination of periodic and harmonic external potentials. In particular, the method of inverse scattering transformation is applied to the GP equation with external potentials. Furthermore, some exact soliton solutions are obtained for the GP equation by using inverse scattering transformation, in which some physically relevant bright solutions are described. The stabilities of the obtained matter-wave solutions are addressed numerically such that some stable solutions are found, and some solitons can be stable in a wide region. These results may raise the possibility of relative experiments and potential applications.     

一类复合材料焊接的界面裂纹问题

利用复变方法和解析函数边值问题的基本理论，研究一类复合材料焊接线上出现裂纹的平面弹性基本问题，笔者通过适当的函数分解和积分变换，将寻找复应力函数的问题转化为求解一正而型奇异积分方程，并借助积分方程理论给出了方程的求解方法。     

Application of rational expansion method for stochastic differential equations

By means of the Hermite transformation, a new general ansätz and the symbolic computation system Maple, we apply the Riccati equation rational expansion method [24] to uniformly construct a series of stochastic non-traveling wave solutions for stochastic differential equations. To illustrate the effectiveness of our method, we take the stochastic mKdV equation as an example, and successfully construct some new and more general solutions. The method can also be applied to solve other nonlinear stochastic differential equation or equations.     

Darboux transformation for the nonstationary Schrödinger equation

The Lax representation for the nonstationary Schrödinger equation with a rather arbitrary potential is proposed. Some examples of construction of exact solutions are given by means of the Darboux transformation method. Bibliography: 11 titles.     

Group theoretic approach for solving the problem of diffusion of a drug through a thin membrane

The transformation group theoretic approach is applied to study the diffusion process of a drug through a skin-like membrane which tends to partially absorb the drug. Two cases are considered for the diffusion coefficient. The application of one parameter group reduces the number of independent variables by one, and consequently the partial differential equation governing the diffusion process with the boundary and initial conditions is transformed into an ordinary differential equation with the corresponding conditions. The obtained differential equation is solved numerically using the shooting method, and the results are illustrated graphically and in tables.     

From the AKNS system to the matrix Schrödinger equation with vanishing potentials: Direct and inverse problems

We relate the scattering theory of the focusing AKNS system with vanishing boundary conditions to that of the matrix Schrödinger equation. The corresponding Miura transformation, which allows this connection, converts the focusing matrix nonlinear Schrödinger (NLS) equation into a new nonlocal integrable equation. We apply the matrix triplet method to derive the multisoliton solutions of the nonlocal integrable equation, thus proposing a new method to solve the matrix NLS equation.     

齐次平衡法若干新的应用

齐次平衡法是求非线性发展方程孤波解的一种有效方法．该文将以KdV方程为例把齐次平衡法向三个方面拓广应用：1）获得非线性发展方程新的具有更为丰富形式的精确解;2）寻找非线性发展方程的Backlund变换、Lax表示;3）求非线性发展方程的对称性约化和相似解．     

Brusselator反应扩展模型的Auto—Darboux变换和精确解

首先用改进的齐次平衡法,获得了Brusselator反应扩散模型的Auto-Darboux变换（ADT）,基于这个ADT,若干精确解被得到,其中包括一些作者的已知结果,然后,通过用一系列变换,这个模型约化为一个非线性反应扩展方程,再采用sine-consine方法,获得了更多的精确解,其中包括新的孤子解。     

无单元Galerkin法在热传导中的应用

对热传导问题的微分方程采用无单元Galerkin法进行数值求解.首先,将微分方程用Galerkin加权残量法转化为等效的积分形式.然后,先将时间变量看作参数,对空间变量进行离散化,得到方程的半离散形式,接着,对时间采用向后Euler—Galerkin格式进行离散,得到方程的全离散形式最后,编制MATLAB程序,上机计算.列举了两个热传导算例,通过计算说明EFG法适用于热传导问题,且其计算速度快,精确度高、前后处理也十分方便,是一种具有潜力的温度场数值计算的新方法.